The amazing thing is, if you have an equation “solved”, you only need 2 points on the curve to compute.
Let’s use the discrete equation:
\[N_{t+y} = \lambda^y N_t\] \[1000 = 60 \times \lambda^{40}\] \[16.7 = \lambda^{40}\] \[2.81 = 40 \times \log (\lambda)\] \[0.07025 = \log (\lambda)\] \[\lambda = \exp(0.07025) = 1.0728\]
i.e. population increase about \[7.28\%/year\].
Estimating some rates … continuous

Let’s use the discrete equation:
\[1000 = 60 \times e^{40\,r}\] \[16.7 = e^{40 \, r}\] \[2.81 = 40 \times r\] \[r = 0.07025\] Intrinsic growth rate = 0.07025.
Washington sea otter fit to data
How long does it take for a population to double?
solve for \(t_d\)
Continuous:
\[2N = N e^{rt_d}\] \[\log(2) = rt_d\] \[t_d = \log(2)/r\]
Discrete:
\[2N = N \lambda^{t_d}\] \[\log(2) = \log(\lambda) t_d\] \[t_d = \log(2)/\log(\lambda)\]
Sea otters: ~10 years.
Sea otter references
- J. A. Estes, J. F. Palmisano. 1974. Sea otters: Their role in structuring nearshore communities. Science 185, 1058–1060.
- Loshbaugh S. 2021. Sea Otters and the Maritime Fur Trade. In: Davis R.W., Pagano A.M. (eds) Ethology and Behavioral Ecology of Sea Otters and Polar Bears. Ethology and Behavioral Ecology of Marine Mammals.
- Gilkinson, A.K., Pearson, H.C., Weltz, F. and Davis, R.W., 2007. Photo‐identification of sea otters using nose scars. The Journal of Wildlife Management, 71(6), pp.2045-2051.
- Veltre, D.W. “Unangax̂: Coastal People of Far Southwestern Alask,”